Resilient Predictive Control Coupled with a Worst-Case Scenario Approach for a Distributed-Generation-Rich Power Distribution Grid
Abstract
:1. Introduction
2. Case Study
2.1. Low-Voltage Power Distribution Grid
2.2. Flexible Assets
2.2.1. Biogas Plant
2.2.2. Water Tower
2.3. PV Power Generation Inferred from Global Horizontal Irradiance
2.4. Initial Operation Strategy
3. Intraday Forecasting of Stochastic Quantities
3.1. Introduction
3.2. Gaussian Process Regression
3.3. Data Description and Kernel Compositions
- Power grid loadHerein, the period of the data is 24 h and the structure of translates the daily periodic pattern. is used to capture the long-term trend of the data, namely the seasonal tendencies present in the power grid load. is used to capture the intraday variations observed in the data, namely intraday fluctuations due to behavioural patterns of end-users.
- Water demandmodels the daily periodic shape of the data and is used to fit the data’s intraday fluctuations. A brief kernel study leads to the conclusion that the simple addition of a squared exponential kernel is sufficient to have satisfactory results without adding too much computational burden to the model.
- Global horizontal irradiancemodels the daily periodic shape of the data and captures the intraday fluctuations present in these data. A thorough study has been conducted in [47] to determine the most suitable kernel composition for intraday GHI forecasting.
3.4. Forecasting Results
- The normalized root mean square error (nRMSE), expressed as follows:
- The coverage width-based criterion () [50], defined as a combination of two different criteria, i.e., the prediction interval normalized average width () and the prediction interval coverage probability ().The criterion allows the surface area of the confidence intervals associated with the forecasts to be assessed:The criterion informs on the probability that measurements would fall within the confidence interval:The coverage width-based criterion () is then defined as follows:
4. Model-Based Predictive Control Strategy
4.1. Inner-Workings of the Smart Management Scheme
- Data acquisition: measurements are taken of stored biogas volume and stored water volume and are injected into the predictive controller in order to update the system’s state. GHI and grid load are measured and water demand is inferred from measurements of water volume and incoming flow rate . These values are injected into the forecast module.
- Forecast module: measured values of the controller’s stochastic inputs (GHI, grid load, and water demand inferred from incoming flow rate) are added to the sliding databases of respective GPR models, which are then used to update the models’ parameters. Afterwards, the module produces updated forecasts of all three stochastic quantities over the forecast horizon, along with their respective confidence intervals. Lastly, GHI values and their corresponding confidence intervals are converted into PV power generation ones.
- Worst-case scenario: phase 1 of the controller’s decision-making process consists in determining the stochastic input values corresponding to the worst-case scenario of the following time slot, in terms of constraint violation. This bloc receives measurements of biogas volume and water volume, GPR forecasts of PV power generation, grid load, and water demand for the following time slot, and confidence intervals of GPR forecasts for the following time slot.
- Reduction of power supply/demand unbalance: phase 2 of the controller’s decision-making process consists in determining the flexible assets’ optimal setpoints. This bloc receives GPR forecasts of PV power generation, grid load, and water demand over the entire forecast horizon, as well as worst-case scenario input values of the following time slot, produced by phase 1.An optimisation algorithm searches for optimal setpoints of biogas plant power generation and water tower power consumption. It does so based on the optimisation problem defined in Section 4.2 and using the low-voltage grid model based on Kirchhoff laws to evaluate various constraints.
- Implementation of flexible assets’ setpoints: optimal setpoints of flexible assets are produced, the first step of which are implemented by the biogas plant and the water tower.
4.2. Main Optimisation Problem
- Biogas plant power bounds
- Switch time bounds
- Biogas volume constraints
- Water volume constraints
- Voltage constraints
4.3. The Worst-Case Scenario Approach
5. Results and Analysis
5.1. Evaluation Metrics
- Final value of the objective function: its square root () represents the cumulative gap between power supply and demand within the power distribution grid during the simulated week.
- Computational complexity : it is quantified by the mean number of objective function evaluations per window, weighted by its size. The number of objective function evaluations is provided as an output argument of the optimisation function “fmincon” in Matlab.
- Mean deviation from the forecasted values: let , , and be vectors grouping one-step-ahead forecasts (herein, 10-min forecasts) of PV power generation, grid load, and water demand, respectively, during the simulation period. This evaluation metric represents the mean deviation of stochastic input values from the ones forecasted at a one-step-ahead forecast horizon. , which is the mean deviation of PV power generation values () from the ones forecasted at a one-step-ahead forecast horizon (in ), is given by:, which is the mean deviation of grid load values () from the ones forecasted at a one-step-ahead forecast horizon (in ), is given by:, which is the mean deviation of water demand values () from the ones forecasted at a one-step-ahead forecast horizon (in ), is given by:
- Instances of voltage overshooting : in cases where the main optimisation problem has no feasible solution, overvoltage or undervoltage may occur in the power distribution grid. These are the instances that the present paper studies and attempts to reduce. This metric records the percentage of these instances during the simulation period.
- Surface area of voltage overshooting : this metric is complementary to the number of instances of voltage constraint violation. It represents the total surface area of voltage overshooting past the prescribed lower and upper voltage levels in the power distribution grid, during the simulated period. It is measured in volts and is calculated as follows:Note that voltage values across the power distribution grid are linked to the values of through Kirchhoff laws.
- Average voltage overshooting per time step : it corresponds to the mean of maximum voltage overshooting over the number of instances at which overshooting is observed during the simulation period. It is defined as follows:
5.2. Impact of Forecasting Errors on MPC Performance
5.3. Contribution of the Worst-Case Approach
- Case 1. The initial case where no optimisation is carried out. The biogas plant has a constant power generation output. The water tower is subject to an ON/OFF controller, which activates its pump when a low-level sensor is triggered and deactivates it when a high-level sensor is triggered;
- Case 2. The predictive control strategy described in Section 4, with GPR forecasts of the PV power generation, grid load, and water demand used;
- Case 3. The amended control strategy proposed in this paper, based on the addition of the min–max problem to anticipate the worst-case scenario within the forecasts’ confidence intervals in terms of constraint violation.
5.4. Discussion
6. Conclusions and Prospects
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Abbreviations
ADEME | French agency for ecological transition |
ANN | Artificial neural network |
CWC | Coverage width-based criterion |
DSM | Demand side management |
GHI | Global horizontal irradiance |
GP | Gaussian process |
GPR | Gaussian process regression |
LHV | Low heating value of the stored biogas |
LSTM | Long short term memory |
LV | Low voltage |
MAS | Multi-agent systems |
MINLP | Mixed-integer nonlinear programming |
MPC | Model-based predictive control |
MV/LV | Medium-voltage/low-voltage |
nRMSE | Normalized root mean square error |
OPF | Optimal power flow |
PICP | Prediction interval coverage probability |
PINAW | Prediction interval normalized average width |
PROMES-CNRS | Processes, materials and solar energy |
PV | Solar photovoltaics |
RQ | Rational quadratic kernel (Gaussian process regression) |
SE | Squared exponential kernel (Gaussian process regression) |
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Evaluation Metric | Case 1 | Case 2 | Case 3 | ||
---|---|---|---|---|---|
4-h | 10-h | 4-h | 10-h | ||
() | 10,035 | 8984 | 8700 | 8966 | 8648 |
(–) | – | 40,872 | 385,320 | 39,700 | 358,990 |
() | – | – | – | 16.26 | 19.41 |
() | – | – | – | 5.56 | 6.73 |
() | – | – | – | −4.24 × 10−15 | −6.41 × 10−16 |
(%) | 23.71 | 5.36 | 6.85 | 4.96 | 6.35 |
() | 4371.4 | 1632.7 | 1464.6 | 1464.5 | 1176.8 |
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Dkhili, N.; Eynard, J.; Thil, S.; Grieu, S. Resilient Predictive Control Coupled with a Worst-Case Scenario Approach for a Distributed-Generation-Rich Power Distribution Grid. Clean Technol. 2021, 3, 629-655. https://doi.org/10.3390/cleantechnol3030038
Dkhili N, Eynard J, Thil S, Grieu S. Resilient Predictive Control Coupled with a Worst-Case Scenario Approach for a Distributed-Generation-Rich Power Distribution Grid. Clean Technologies. 2021; 3(3):629-655. https://doi.org/10.3390/cleantechnol3030038
Chicago/Turabian StyleDkhili, Nouha, Julien Eynard, Stéphane Thil, and Stéphane Grieu. 2021. "Resilient Predictive Control Coupled with a Worst-Case Scenario Approach for a Distributed-Generation-Rich Power Distribution Grid" Clean Technologies 3, no. 3: 629-655. https://doi.org/10.3390/cleantechnol3030038